Calculus of Real and Complex Variables

11.1 Sard’s Lemma and Approximation

First is an easy result about approximation of continuous functions with smooth ones.

Theorem 11.1.1Let Ω be a bounded open set in ℝnand let f ∈ Cc

(ℝn )

. Then there existsg ∈ Cc∞

(ℝn)

with

||g− f||

∞< ε.

Proof:Form g ≡ f ∗ ψn for a mollifier ψn. This will approximate f uniformly on Ω, and will be in
Cc∞

(ℝn )

. ■

Using the Weierstrass approximation theorem, you could also get g to equal a function from G described
in the development of the Fourier transform for all x ∈ Ω. Simply apply Theorem 2.8.11 to the functions in
G.

Applying this result to the components of a vector valued function yields the following
corollary.

Corollary 11.1.2If f ∈ C

(-- )
Ω;ℝn

for Ω a bounded subset of ℝn, then for all ε > 0, there existsg ∈ C∞

(-- )
Ω;ℝn

such that

||g − f||∞ < ε.

The following is Sard’s lemma. It is important in defining the degree. This was shown earlier in the
presentation of change of variables formulas. See Lemma 8.3.4.